Abstract FR:

This thesis is focused in the framework of biological dynamical systems. Its main objective lies in the fact that from a mathematical formal study it is possible to infer and answer interesting biological questions. The fist problem addressed are the Thomas conjectures, which set that necessary condition for the existence of attractive cycle (resp. Multistability) is the presence of negative circuits (resp. Positive) in the regulatory graph. We begin proving a series of lemmas in order to give conditions over the transition graph, along with a general sign formula. This makes possible to give an alternative proof to a theorem of Remy et al. About the second Thomas conjecture. In the second case, we find conditions for the existence of cycles in general. Moreover, we define the “Extended transition graph”, which contains not only information on the dynamics but also about the structure of the regulatory graph. In the second problem we show that the method of desynchronization proposed by E. Pécou can be simulated in a numerical stable way. We apply with success the algorithm to the Goodwin model with positive and negative regulation, showing theoretically how to induce periodic behaviour by adding a new equation. The chaos induction of Shilnikov or Lorenz type, depending on the nature of the eigenvalues, is shown by the construction of homoclinic orbits and the sensitivity to initial conditions. Finally, in the last chapter, we propose a mathematical model accounting for the processes of uptake, efflux, storage and traffic of transition heavy metals Cu, Zn, Mn and Fe in Halobacterium NRC-1 based on the framework of differential equations and the power law formalism. We prove in a formal way that the systems present stable stationary states. Additionally, we derive monotonicity conditions for the existence of global steady state responses, independently of the choice of the parameters. Together with the theoretical results, we develop several simulations to answer biological questions abut growth and death at high metal concentrations and the cellular response to the stress produced by alternate and successive increments of metals.